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Math 166 - Exam 1 Reyiew

NOTE: For reriews of the other sections on Etam l, rctrer to the frst page of WIR #I and #2.

Secrion 1 5 - Rules for hobabilitv

. El€mentary Rules for Probability - For any event E in a sample space .t, we have

Rule l : 0<P(E)<1

Rule2: Pl$: I

Rule3: PfO) :0

. Union Rul€ for Probability - If E and F are any two everts of a sample space ,t, thenP(Er F): P(E)+ P(F) P(ErF).

. Union Rule for Mufualy Exclusive Ev€nts - If E and F are mutually exclusive events, thenP(Eu F) : P(E) + P(F).

. I f Et, E2,. . . ,E^ arc mutual ly exclusive events, lhen P(tr U E2 U . .UE,):P(E1)+P(E2)+. .+P(E").

. Complem€nt Rule for Probability - If E is an event of an experiment and E denotes the event that t does notoccur, lhen P(E') : t - P(E) and P(E) : I - P(E ).

. Odds - Odds are another way of representing the probability of an event and are given as a mtio of the number oftimes the event will occur to the number of times the event will not occur in the long run. Mathematically, the odds

in frvor of an event E are define to be a ratio of P(E) to P(E'): odds in favor of E : A;.

NOTE: we reduce

PG\the ratio

6 !o lowest terms.

;. and then we say that the odds in favor of E te ato b or a: b.

. Obtaining Probability ft,om Odds - ff the odds ir favor of an event t occurring are d to ,, then the probability ofc occurTrns rs r lz ) :- a+b

Section 1.6 - Conditional Probability

. Conditional Pmbability - the probability of an event occuting given that another event has alrcady occurred.

. We denote '1he probability of the event A given that the event B has already occuned" by P(A B) .

. Conditional Prob&bility of an Event - If A and B are events in an experinent and P(B) I 0, fien the conditional

probability that the event A will occur given that the event I has already occuned is PIA Al : P(A,n,A).

P(BI

. Independent Events - Two events A and B are independent if the outcome of one does not affect the outcome ofthe other

. If A and B are independent events, then P(A a) : P(A) and P(8 A) : P(B).

. Ind€p€ndent Events Theorem - Let A and B be two events with P(A) > 0 and P(8) > 0. Then A and I areindependent if and only it P(ArB): P(A)P(B).

. NOTE:To determine if two events are independent, you must compute the tlree Fobabilities P(A nt), P(A), andP(B) separuteLy, and then substitute lhese thfte numbers into the equation P(A n 8) : P(A)P('). ff the equalityholds, then A and B are independent. If after substituting you 6nd that P(A n B) I P(A)P(B), then A and , areNC'I independent (i.e., A and B are dependent and somehow afiect each other).

Page I

Md rft Fi' 03 ox!,h. Rdq

. IndependenceofMoreThanT\yoEvents-f fEt,E2,. . . ,E,areindependentevents,then

P(ErnEzn nE,): P(81)P(E2) P(E")

Seation 1.7 - Baves' Theorem

. There is a huge formula in the book for Bayes' Theorem, but you don't have to memorize itl Just make sure youknow the conditional probability formula P(,4ID) : ffi, and know tro* to use a tree diagram (or Venn diagramin some cases) to find lhese probabiliries.

P^ge 2

1. Consider the propositions

p: Bob will have a hamburger for lunch.4: Bob will have pizza for lunch.r: Fred will have a hamburger for lunch.

(a) write the pmposition r A (p y 4) in wods.

FTF

tr

TF

T

{

FFF1

f { -

( t

TFTr

Ir{

FF

Frad oi\\ h^\)c d, harn.tt.vgcr- k Luntt+ ,awl ?vb .:\l\ z)I\ar hl, ,<-a-M'abur1f.r 0r l\tu<, tx Lq n*,. (bv+ nsS- rrorl_1.(b) Write the proposition (4V - p) A r in words.

bot,r't\ hr. u< prtu- kr lar.ttLt,0 r ht,,o'{t ^01

Mrt ohtrnbu-rgt-f fti. lt^^c(,L,a{,.kcd

-\.r v,4.'< a ta u.ls6rq(/ k* tuun.

tcr Wrile'tfid proposition 'Bob and Fred witl both have a h)inburger for lunch. or Bob willhave piz,,a tor lunch.'symbolically. / \

(PA()r 'V(d) Write the proposition "Bob will not have a hambuger or pizza for lunch, but Fred will have a hamburger for

lunch," symbolically.I

- (pv y) A r2. write a truti Lable foreach ot tie lollowing.

(a) -(-pvq)v(p^-q)

3 :t tg _f vt "Oev,h) pn^,b -(_gvt) V (p/l-r.)

F{

F(

l-

TtrF

(b) - r^ (4v - p)

Mdi r# d' 43 otue. Rliey Page 3

F

T

rFF-1-

F-r

v-P

Ft-T.'r'

-l-

TFFFgTI tserkffT€FrTTTFF{r

kTT\-

FTT

(

(

7TT{

F

r

Pmbl€ms 3,4 6, and E are

True or False. U : {0, I , 2, 3

courtesy ofJoe Kahlig.

,4,5,6,7,8,9) andA : {0,1,2,3,4,51* 'AT (F , '.r l;1- \'-./

r6 oe a

(T)F I1.2.31cAT \F ) 2CA

r (O nr.q.r = s

; " ,P; t ' : ' *eeY@ tot=o

n({3,4}) = 2n(o) : t3eAc0:0

A= Ia,b,c j

(a) List all subsets ofA.

fi , 1^3,18, Lc3, La,L&, I",cJ, 1b,cJ, fu,,v,c\(b) List all ofthe proper subsets ofA.

i l i \4t at tl ' a+u(( 1a,b,4

(c) Give an example of two subsets of A that are disjoift. If this is not possible, then explain why.

z ) ,1t7 | ' 'n t r l Jfu\ a^a lD\ a '^! d t5)D\^? 1t^u t^3 ( \ tb l = F

5. Shade the part of the Venn diagram that is rcpresented by

(a) (^cu 8) n (cuA)

^&r14*{

it:',if,J"-@ia

Mri 16 h[ 1M Osedr tu&y

(b) (, uc)nAc

l1- l ,a,J ' t -

^^ ^. , , i , r . . \ bvc- t t lW)2to

' '' ' ' ,G;i;;;|, ^E

^.6-A"-GW(ut

Page 4

r,(-

6.U: lo, t ,2,3,4,5,6,7,8,9j ,A:{r ,3,5,7,9},B:{1,2,4,7,8},andC:{2,4,6,8}.Computethefol lowins.

(a) (A nB) uC

av--lt,l\

(c) An (BuC)c

g,tc. l to,+, ' t , t ,6\

(0,t4" ,10,2 ,t ,4J

(eno)v c -- f t,t, z,+,a,t\

ft n (&uc)'-- 1t,s,n!

@) Ac nB

A,.10,L,4,t,,5JA' to --{t,,1,lJ

7. l.€t u be the set of all A&M students. Ilt

A : {r € U r owns an automobite}D: {_x € Ulr lives in a dorm on campus}F : tr € Ulr is a fteshmanl

(a) Descdbe the s€t (A n D) u 1.' in words.

(b) Use set notation (n, U, ") to $.rite the set of all A&M students who are freshmen living on campus in a dormbut do not own an automobile,

,I"!zf qaJ.I lttll:ht-lu+s vrX, o Wla Ovl a""r arl*frrtobiie

altle \;.^dJh^ 6r.-\.AA.PtL5) or u.'Lo a.^e {\0+ fn*Lwn

FAD OAC

Mzr rm Fjr ,4 @Hdb bry

8.

Page 5

In a survey of 300 high school senio$:I2O h^d not rcad Macbeth bfi had rcad As you Likt It or Romeo and Juliet.6l h^d rcAd As you Like lt b^tt not Romeo ond Juliet.15 h^d rcAA Macbeth al:,d As you Like It.14h^d rcAA As You Like It and Romeo and Juliet.t had rcaA Macbeth and Romeo a.nd Juliet.5 had read Mocbeth and Romeo and Juliet t'/ut not As you Like It.40 had read only Macbeth.

Ie|M = Macbeth, R = Romeo and Juliet, ^nd

A= As you Like It.

(a) Fill in a Venn diagam illustrating the above information.

l1-o -5-D -t o

(b) How many students read exactly one of these books?

4u t too ,.5n =16l

(c) How many students did i$ rcad Romeo a.nd luliet?

(d) How many srudents read Macbelh or As you LiAe lt and also rcad Roneo and Juliet'!

4o {-il rs-o + ao =F4

5++ f t0=lo(e) computen(MU(AtrnA)): \\+ 50 f 40 k 4 | 5'nA

(cAA - l,e

^^- f , ! '4,c)1,

Rt- (,u,",r. '(

f t - al lo,d'eJ V\U (KAA) - b,"- , f 3,c

h6 G6 tun'Oklkridry

9. Find r(A n B) i fn(a) :8, r(a) :9, and n(A uB): 14.

n(AuB)-- n(A) + n( 6) - ^t A \g)

t4 -- t +1 - ^(A^a\l3 = n/Ano)l

10. A bag contains I pennies, a nickel. and two dimes. Two coins are selected at random {iom the bag and the monetaryvalue of the coins (in cents) is recoded.

(a) wlat is the sample space ofthis experiment?

^ 1. aa'- l&t tot l l t l t , ln7

'J

O) Write the event t lhat the monetary value of the coins is less than I I cents.

1^,nY'-= Ld'\ !J

(c) wlite the event f that the nickel is drawn.

t4€ =1 ln. 151

J

(d) Are the events t and F mutually exclusive? Support yout answer.

Sinus E Otr , l t"\ *Q,-!4-$.U.14\ J(

Page 6

(P - /?\ . r"P\ , '

dd. ',."

Eand € a,u"vw+ {^} lua%.

(e) Wdte the event G that the value of the coins is more than 25 cents.

/ -Z7 lI - - 4 7 - r - t \!) '-1 \ - v-)

I

I 1 . If p is a tru€ statement and q is a false statement, find the truth value of each of the following statements:

tat - n!-q\

T

(b) pv-sIVT

..

(c) - (a^-P)

^ {r 'nr)

= - Ltr\

@hvu

br6Fd@3oq.rbrry PageT

12. La S: {a,b,c,4e, f} be the sample space of ao experimenr with the following Fobability distribution:

ouko" l !

b- , ! ! - : I - , t - . t9**o"o*oToo1 * m(;; ,ao # e- [ -7o'

n

t a 4 z r*^-1r*t t - {o. c. e\. s - {c.././}. and c [b.d I be events of $e ex""..;; J,n;t$ ;,Xi

-lo? *

(a) Fill in tbe missing probabilities in the Fobability distribution above.

iv. P(A n B)

A^blqv. P(Arl --+o I

\ut--lac,gdg!

2! - e --EI40 - 50 L-t-J

((nno= klu I,

+ iu ' q"

(d) Are the events A and C mutually exclusive? Why or why not?

A nc=?c] +V A rvir t",.,Lvurl\.e4a!.u; tt.

,J9l ln

4q

+or 4e

(b) Is this a uniform sample space? Wlry or why not? ,

1Jo. g*rporrra s rr^e ^

rt %iljll 3 !'k I"J.

{cl Fiod each ot lhe following: 2 1 , f;li. p(A) = ?(a>+?G) lp(O--

-qo | {o ,io =l 6 \

ii P(c) -_ ftL)+ pU)

'k 'k &iii. P(rc)

=l-?(b) = l -

M4r6anr@oH&b4

13. One card is drawn at mndom from a standard deck of 52 playing cards. What is the probability that the card is

(a) acrub? C-.uftt ,l,ttt* .11^s-c4rd is aV.*b

Pru>-- {$(b) a face card?

F-arttr,t'LLati l ,etavd ts altce rcrA

fttr)= ffi =(c) a club or a face card?

P(L u F) : elu)+PlF\ - P{o^F) _--vt ,z- 1- l -41_ s> . 5) 5n _

L5a I(d) neither a club nor a face card?

f( c-"nr') '. p((cur).) = | -g(c u r=)- t -U -6

14. Let E and F be two events of an experiment s*,Inu'",u, 3Jrr. ",FPo4.

and p(t ur) :0.62.

(a) Are E and F mutually exclusive events? fttJab $ ^t Pf E n tr)

fiZ-l- lsal

?Lev") -- P(€) +P(tr) -f{Entr)A,\pt - - \As+\,1 - ?tenr>

D,g' 'e(e^F)(b) Find P(t'uF ).

P[L.uF.), Pl€.nO")

'aTqY-v'.Y"'o'b

f -UFc-a'b,c

= l- fllFAF) ; | _a,lz5 ,iO,{tr(c) Find P(EiF ).

(d) Find p(rur"): 0,K+ 0 ,L1-l 0,\V

Page 8

:i*crP(E0O)0, i| i1 @ssible'&r lLP:+ +t"rO euent+ *O acour a.,1l^aaa*q n//.j'. 'ffure{o.rt L a-JF a"l tot vral lv,a-U.l.r al&Luautt .U

0A-o.tz

0,45'a \z

l'0.la - 0.13 -0,?-7

M 16 an ,@3 oHc{Ei R2ruy Page 9

15. A survey was conducted in which 1,000 students at Random University were asked how many hours they arecurrently taking. The results arc given in the lable below Use the table to answer the following questions about arandomly selected student who participated in this survey

F- freshl'u,,rfl- 6o0rh-osto.{osJ - Ju{ idv ' t

N - 5{ rr, 6v5

Classification y or less l0 to 13 14 to 17 18 or more TotalFreshman

SophomoreJuniorSenior

l0l525

!50

13090105i lo

1405580I{J

5203545

285180245290

Total 100 435 360 105 1000

(a) What is the probability that the stualent is a freshmanflre or she is registered for 18 or mote hours?

lL.rt'- k.4t ' , l \^

?(r \u) = 5' a' Ptr*| t+r rt\<kre 4 ftk lg on ",.ryf iur{,(j.

' t , '1. ' . ' . , - i f i r -J l ,dJ ' , t '

l \ ,>,nt ,o t tv,sJ

(b) Wlat is the probability that a@!fs registered for 14-l? hours?

0/ n l r ' . \ \ - E'-{- ' "s rQgst ' r t ' t ht t4 t ' t ivs

| \ \_ l f ! / - AaO<,x.n \ {'l5

(c) what is the probability thar the studenr is a jun;"ri:,: li .l" ['j"_,iltr"Hx,: i?: ,:i."t

o/-< 1 , .f tJ IAUbi= -6r+W =

l - rAr1p6'16-ffi-a 4e rycc

16. A student has two exams in one day. The Fobabiliry that he passes the first exam is 0.9, and the probability thathe pass€s the second exam, is 0.85. If the probability that the student passes at least one of the two exams is 0.97.*gQ"i9)*o

"u"ot(6 p"'iir"nt? fr- p"cgs lsv u*' q-pds.".t O,-/ t t7t"a-

(eiL

Y(AnDt?(A\ftb>((Avl)'t qt

p ([o h\. p (A) + plb) -P( A Ab)0,4t '04r) :16-wAnb)cl'L i (lnslno.f rrvl".qrr'd.lrLt

o,1s i to4)(o'st)

M,h re Fdr M3 oHdkr Rqe Page 10

l'7. I Et A, B, and C be thle€ independent events of an experimenr with P(A) : 0.4, P(A) : 0.75, and P(C) : 0.3.Calculate each of the following.

@) P(Anf) -.' PfA) plec) (. ' . ' ,az lxlaywda*)

I r-urr)(0,,1)ic

=EA

-t-------

Pa)= (o,r)(o tt) + (o.e)(o,t-)=l---uqf J(c) What is the probability that an applicant is apnroved for the jo(i{re or she is unqualified?

P(At0") -tg!l rm k.,)(d) What is the probability that an applicant is unqualified {i he or she is approved for the job?

P/ n. lR) - ?g!D-, !e t . / " p{A)

t t - n 09lb-"-- ! v

l?\

?(qcyc)= ptA,)+ p{c) - ftA,nc1' (r-u..r) + 0,1- (t-0,+)(e.31 =]F?-i.t

(c) P(Blc)

l (b\c) = {lb]l[iru r*ur-.p)

ff - opp;cn^ + ;"Q - n'rt,{.,r,,1 afPt' ro t v

*Os+@ *trbpr*r P(A-iq=iDE_ kL

rbr what is the probabil,o ** * *i'/-r 1;{oort"l*t

18. The personnel manager at a certain company claims that she appmves qualified applicants for a cedain job 85% ofthe time; she rcjects an unqualified person 80% of the time. If 70% of all applicants for this job are qualified, findeach ofthe following. (B iu,dn tr'derydbrb.c"-wn.dridq)

(a) Draw a t ee diagram (with probabilities and notation on all branches) representing the above information.

M'r r6F!'4soHrebq Page 11

19. A ?th gade class was selected and the following information was collected about the 30 students.

4 students have only glasses.12 snrdents have only braces-6 students have glasses and braces. t@'".

Det€rmine whether the event that the st!!d9!Lba!gla!!e! and the event that the student has braces are independent.Justify your answer G- zkuAp aj |,t.. *as:>f,t

g- "ta{a(a\

&ft- , .2,e(Gl'6)?,flcifiP{ f i = -",,'-to 'lZ6X6) '-/

20. The odds that il_udl not snow in College Station next winter are 17 to 2. Wlat is the probability that it will snowin College Station next winter?

P{.; l l toru) -

p (a,n!

Or WnulR U. pro#Ultlry lhat the 6rst marble rs green and the second marble r( bluel

(a) what is the sample space for this experiment? Q-td Vnlblt,

K-l P:ffi^Yi!h'"

dds 11 to ? 'd4 a"

11 t-t'tq' t t. ,. , 11 no +

7\a-o ,i"o r e.o"-i! 3 +i,wt, t til. :,., ,t

'-\t-|..j.

21. An experiment is as follows: One marble is drawn from a bag containing 3 red, 7 blue, and I green marble. Thecolor of the marble selected is observed. If the marble is red, then a fah coin is flipped and the side landing up isobserved. If the mmble is green, then a second marble is drawn ftom lhe bag and the color is observed.

/ \ , r .- ,1\V( qru,nl* anA H^ a* ) = p(qw nt ;p(Vtur 4il-- / +\l+\ -E\

(c) wrire lhe evenr rhar a r,* .-t. "I "1"."'.1oii"i *. *ol"l"!?J

' ' ^Y' --7,b,

3. f{x,*1,LR,i) b) (G/R)r nr,t

')

i l3,*n'u..

G 6^l B a^t r nltpt'tdl*4

Md 16 Fd[M OHdio hq

22. A bag contains 5 pennies,3 nickels, and 7 dimes. A purse contains 4 nickels and 6 dimes. A coin is drawn fromthe bag and dansfened to the purse, but if a nickel is s€lected from the bag, then all 3 of the nickels that were inthe bag are transferred to the purse. A coin is then drawn from the purse. The type of coin drawn from each of thebag and purse is recorded. Wlat is the probabilily rhct p, , 1rt" _y k*n tro,

(a) the transfen€d coir was adime if a nickel was selected from the purse? Nr- t\i Ciel " ,,

n, @ Dr- o, . . t | ' ,f ) -

' t '

U-| ,nl, 'i--i ? t ,' Q::z f, . ycxr,' t'r\". {}tt.t ;t_ n S"<|f,y)' r U, 4N Ny . ;.r i"t)'fr *=:@l1o.+

_ _?o. D1 . .,1 , nt

* ffi*, j- oi3**'" dPlu'1t'to)' 'W

r , \ '

,E vz ) l 'D

'4).->-.-o Artr\?)'-i- ri1, - ,,.--,

Page 12

23- Two cards are drawn at random fiom a standard deck of 52 playing cards. What are the odds that the second carddrawn is a khg given that the first card drawn was a queen?

nrrn\-+ . , - (1. \ ( t \tD, fll:ry rt- N! ? s.j= '- - -==:;?4rL1---- ' fi;;;.-_=--; . ;f (7.)(fr)r(L*)(1) 44"\:+)' ' -21Dr)_ f r - "2,

,aD r lD )B f f i t

: - tk{-- - IZJ.^. t t "1o) both coins are pennies? A*O -

LEZ) ^ t' '' ' '

f1p,nP1)' &)(!'\=6

(c) the coin drawn ftom the bag was a nickel or the coin drawn ftom the purse was a dime?

p(t't,u lr;. P1N) r p(D,\ - P1tu,n D)) -" -i;, ,\= g +[,{")(+;)r{il(it)r(aXpl -(?)(?") '-lB(d) the coin drawn from the nurse is a Wnnf if)he coin drawn from the bag was a nickel?

g (ilutU A.{ r1t Pr,!r'r,ft r,r. i '^'L pu roe)P( l," lN).

l rohth; t ;4a = 4rJ=a

) l

tl

ilds 5- ,\ - t ,.l

-41

rl-1 Uj:t-,

M'h 16 Fdr03 oudrd rMq

24. Thre€ rnarbles are drawn one at a time without rcplacement from a bag containing I I red and 9 blue marbles.

(a) Draw a tree diagram with probabilities onall bmnches for this €xperiment. Ki - f?d w.a rLl( t$ Lhc

(, .

( t i . Wuwtbi , t1a; t t ,i ' - 1 ,7,1

61

*...i:#?i

F(t, n cr' rln')+ p(c:Ac, nr'.r") +P10lnc,'nru)(q 12)(r -o,ut ri | -u,\ + (r-olr)0 s[t-u,) * { t - o,rQ[r-o,od) lo'

-:--,--------'7

Ul\sH- ' -

Page 13

n!-

-z--=rE

O) What is the probabiliry that the third marble setectea is ltue? $

p & it = P, )P )( ) r (),,)(ft )(& ) + (*,)( :) ( *) . i*,Xft X r -l

(c) Wlat is the probability that th€ third marble selected is blue if it is known lhat at least one of the first twomarbles \€lecled was Ied?

Q/A-l . , ,00, . { (N r?d ,^t t 1d,0.) \ ) - ?U>z-Aa'u ' ' t

r 'o l ' ' r dJt t -1t

I ( ,JI{ ' Ybr 'L 'c ' i t i ' "e \ t ' ' ! ^1} ' }dta") ' )L-r(*)(3<> '"d't*"t,[)GlrkX%) " (a)(h)

=' ]# =l* \25. A buildhg on campus has three vending machhis:'t'ivb Coke ifihines xnd a $ack machine- Based on the model

of the machines, the nrst Coke machine has a 12% chance of breaking down in a particular we€k, and the secondCoke machine has a 4% chance of breaking down in a particular we€k. The snack machine has a 10% chance ofbreaking down in a particular week. Assuming independence, find the probability that exactly one machine breaksdown A - 0r,r*, Jrr, i" t t+ ( i l tc {Aa(n,to i luyS J.d-lor\

U\2 rLlvr" ! ' - -

fr r , , ' " a.v\a

ni - . ' ^ t tLW[h,u

r ' ' 't \

(1turll1 \ Wtnlrs lr*) -

Cir n rvt,r i i ! ' i r - ' .^V

.-1a^)tt{tt fie -i,'40ffrwlit

-